NCERT Class 12 · Ch 7 Exercise 7.6 · Q22
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.6 Question 22
Integrate sin⁻¹(2x/(1+x²)).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: For |x|≤1: 2x tan⁻¹x−log(1+x²)+C. For |x|≥1: π|x|−2x tan⁻¹x+log(1+x²)−2log 2+C.
- Principal branches give 2tan⁻¹x inside [−1,1], π−2tan⁻¹x for x≥1, and −π−2tan⁻¹x for x≤−1.
- Integrate each branch by parts using ∫tan⁻¹x dx=x tan⁻¹x−(1/2)log(1+x²).
- The outside constant −2log 2 makes the pieces agree at x=±1. Thus this joined primitive is valid on all R; retain a common arbitrary constant C.
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Evaluate \(\int e^{2 x} \sin{\left(x \right)}\, dx\).
Next Question →Evaluate \(\int x^{2} e^{x^{3}}\, dx\).
All Questions in Integrals — Exercise 7.6
Related Questions in Integrals
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int x^2e^{x^3}\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int e^x(1+x)\sec^2(xe^x)\,dx\) on an interval where defined.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int e^x(\cos x-\sin x)\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int dx/(\sin^2x\cos^2x)\) on an interval avoiding its poles.
Class 12 Integrals · Mcq Single Correct
If \(\int\frac{3e^x-5e^{-x}}{4e^x+5e^{-x}}dx=ax+b\ln|4e^x+5e^{-x}|+C\), find a,b.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).